First, it's far from certain that she could actually help Obama in Missouri. She lost a statewide election for governor, and in 2006 -- one of the best years for Democrats in recent history -- she just barely squeaked by Jim Talent, who hadn't even had the opportunity to serve a full six-year term. She did so by trying to seem conservative enough on social issues to hold down the out- and down-state vote, while racking up big margins in the cities (with the help of her buddies -- and Barack's -- at ACORN.
As for experience, well, she's been in the Senate less than two years.
She obviously wants the job, but it seems pretty certain now, at least, that she's not going to get it -- and not just because she's sort of campaigned for it.
As Amanda points out below, McCaskill has made remarks that sound like she's accusing everyone who doesn't support Barack of racism. Sure, she may just have overcompensated in her efforts to prove that she could be a vice-presidential attack dog as tough as the next guy, but she's just undermined her own chances.
Over at the Atlantic, lefty Matt Yglesias gets why all this is a problem: "[T]rying to sniff out racial subtexts in these kind of things [like the McCain "celebrity" ad] strikes me as overwhelmingly likely to prove problematic. People really don't like to be called racists."